Question

the length of one side of △ptv is given. use the relationship between the sides of a 30°-60°-90° triangle to find the lengths of the other two sides.
if your answer is (5sqrt{2}), please type \5sqrt2\ - no space.
given (vt = 6), complete the table and find the missing sides (radical form)

Explanation:

Step1: Match side to 30-60-90 rule

In a 30-60-90 triangle, the side opposite $60^\circ$ is $x\sqrt{3}$. Here, $VT = 6$ (opposite $\angle P = 60^\circ$), so:
$$x\sqrt{3} = 6$$

Step2: Solve for $x$

Isolate $x$ by dividing both sides by $\sqrt{3}$, then rationalize:
$$x = \frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$$

Step3: Find $PT$ (opposite $30^\circ$)

The side opposite $30^\circ$ is $x$, so:
$$PT = x = 2\sqrt{3}$$

Step4: Find $PV$ (hypotenuse, $2x$)

The hypotenuse is twice the shorter leg:
$$PV = 2x = 2\times2\sqrt{3} = 4\sqrt{3}$$

Step5: Fill table values

Map $x, x\sqrt{3}, 2x$ to the angles:

• $30^\circ$ side: $x = 2\sqrt{3}$
• $60^\circ$ side: $x\sqrt{3} = 6$
• $90^\circ$ side (hypotenuse): $2x = 4\sqrt{3}$

Answer:

Table:

$30^\circ$	$60^\circ$	$90^\circ$

Missing Sides:

$PT = 2\sqrt{3}$
$PV = 4\sqrt{3}$